Find a polynomial with integer coefficients that satisfies the given conditions. R has degree 4 and zeros 3 − 2i and 2, with 2 a zero of multiplicity 2. R(x)= ?

Answer 1

The key to problems like this is noting that, as the solution has integer coefficients, for each complex root, you must have its complex conjugate, and each irrational root must have its conjugate as well. After multiplying these pairs, if some of the terms in the expression are non-integral, you have to multiply the x and the other terms by the least common multiple of the denominators.

Then, for this term, R(x) = (x-2)² (x - 3 - 2i)(x - 3 + 2i)

I will take you one additional step (x-3-2i)(x-3+2i) = (x-3)² - (2i)² = (x-3)² + 4 = x² - 6x + 13

Then, the solution is (x-2)² (x² - 6x + 11) If you need to write out the entire polynomial, I leave it to you to expand the first term and then multiply the two terms together.